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  • Help me answer: Population of a certain country is know to increase at rate propotional to the number of people presently living in the country. If after two years, population has doubled and after four years population is 20,000, then initially popu

Population of a certain country is know to increase at rate propotional to the number of people presently living in the country. If after two years, population has doubled and after four years population is 20,000, then initially population was

  • Option 1)

    5,000

  • Option 2)

    4,000

  • Option 3)

    3,000

  • Option 4)

    2,000

 

Answers (1)

best_answer

As we have learnt,

 

Growth and Decay Problems -

We assume that rate of change of amount of substance is proportional to the amount of substance present

-

 

 Let at time t, and N be the number of population at that time, So

 \frac{\mathrm d N}{\mathrm dx} = kN \\*\Rightarrow \frac{\mathrm d N }{N} = kdt

So, on integrating we get,

\ln N = kt + c

Let intially population is N_0 at t = 0

\Rightarrow c = \ln N_0 \Rightarrow \ln N - \ln N_0 = kt\Rightarrow \frac{N}{N_0} = e^{kt}\Rightarrow N = N_0 e^{kt}

It is given, after two years, population is doubled, So 

2N_0 = N_0 e^{k\cdot 2}\Rightarrow e^{2k} = 2\Rightarrow 2k = \ln 2\Rightarrow k = \frac{1}{2}\ln 2 \\*\therefore N = N_0 \cdot e^{\frac{t\ln 2}{2}}

Its given, after four years population is 20,000, So

20000 = N_0\cdot e^{2\ln 2} \Rightarrow 4 N_0 = 20000 \Rightarrow N_0 = 5000

 


Option 1)

5,000

Option 2)

4,000

Option 3)

3,000

Option 4)

2,000

Posted by

Himanshu

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